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Theorem prnzg 3483
Description: A pair containing a set is not empty. (Contributed by FL, 19-Sep-2011.)
Assertion
Ref Expression
prnzg (A 𝑉 → {A, B} ≠ ∅)

Proof of Theorem prnzg
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 preq1 3438 . . 3 (x = A → {x, B} = {A, B})
21neeq1d 2218 . 2 (x = A → ({x, B} ≠ ∅ ↔ {A, B} ≠ ∅))
3 vex 2554 . . 3 x V
43prnz 3481 . 2 {x, B} ≠ ∅
52, 4vtoclg 2607 1 (A 𝑉 → {A, B} ≠ ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390  wne 2201  c0 3218  {cpr 3368
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-v 2553  df-dif 2914  df-un 2916  df-nul 3219  df-sn 3373  df-pr 3374
This theorem is referenced by:  0nelop  3976
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